JOURNAL
OF COMBINATORIAL
THEORY,
Series B 47, 153-161
Combinatorial
Reconstruction
N. ALON*, Y. CARO, I. KRASIKOV, Raymond
(1989)
Problems AND Y. RODITTY
School of Mathematical Sciences, and Beverley Sackler Faculty of Exact Tel-Aviv University, Tel-Aviv, Israel Communicated Received
Sciences,
by the Editors February
4, 1987
A general technique for tackling various reconstruction problems and applied to some old and some new instances of such problems.
is presented 0 1989 Academic
Press, Inc.
1. INTRODUCTION Let X be a (finite or infinite) set and let G be a (finite or infinite) group of automorphisms of X. Thus G acts on X and for every g E G the sequence is a permutation of X. For every subset Y of X and every g E G, k&x let g Y be the set of all elements gy, for y E Y. Clearly 1 g Y 1= 1 Y 1 for every finite Y, and this defines an action of the group G on the power set of X. The orbit of a subset Y of X is, as usual, the set YG = {g Y: g E G}. We say that two subsets Y and 2 of X are G-equivalent iff there is an element g of the group G mapping Y into 2, i.e., iff ZE p. Let R be a set of representatives of these orbits, i.e., a family consisting of a unique member of each orbit. For an integer k 2 1 and for a subset Y of X of cardinality m > k, the k-deck of Y is the function d = d,,: R -+ (0, 1,2, ....} where for each YE R, d(r) is the number of subsets of cardinality m - k of Y which are G-equivalent to r. Notice that CrER d(r) = (7) and that if Y and 2 are G-equivalent then 4,/c = dZ,k, i.e., the k-deck of Y is determined by the orbit of Y under the action of G. Several problems in combinatorics and in graph theory, known as reconstruction problems, are special cases of the following general problem: Problem 1.1. Given a set X, a group G of automorphisms * Research grant.
supported
in part
by an
of X, and
Allon Fellowship and by a Bat Sheva de Rothschild
153 00958956/89 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
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ET AL.
two integers m, k (k cm), decide if any subset Y of cardinality reconstructible (up to G-equivalence) from its k-deck.
m of X is
The best known instance of this problem is the edge-reconstruction conjecture [H]. In this problem X is the set of all (i) edges of a complete graph K, on n labelled vertices and G is the group of all permutations of X induced by permuting the vertices of K,. Every subset Y of X is a graph on n vertices, and its orbit is the set of all graphs isomorphic to it. Here k = 1, and the l-deck is just the set of isomorphism types of all edge-deleted subgraphs of Y, each appearing according to its multiplicity. The edgereconstruction conjecture asserts that for k = 1, m > 4, and X, G as above the answer to the decision Problem 1.1 is “yes,” i.e., every graph with m > 4 edges can be reconstructed from its edge-deleted subgraphs. The well-known result of Muller [M], extending that of Lovasz [L], is that this is true for m > 1 + log,(n!). More generally, it is shown in [GKR] that for X and G as above, for k >/ 1, and for m > k + log,(n! ) or for m > I[ (‘;) + k] the answer is also “yes.” Here we consider the general Problem 1.1. For a method that works in various other situations see [KRl, KR2]. Our results have several applications to various old and new reconstruction problems. In particular, we generalize a result of Nash-Williams on edge reconstruction of graphs to more general combinatorial structures. The paper is organized as follows. In Section 2 we state and prove our main results concerning Problem 1.1. In Section 3 we consider some applications of these results to several reconstruction problems. In particular, we show that any colored cycle with m > 5 vertices of color 1 and n -m vertices of color 2 is uniquely determined by the set of the m (unlabelled) colored cycles obtained from it by replacing, in all possible ways, a vertex of color 1 by a vertex of color 2. We also obtain several new results for the edge-reconstruction conjecture (for graphs and hypergraphs) and for the vertex-reconstruction conjecture. We further discuss some geometrical reconstruction problems. In particular we show that every polygon with m > 8 vertices in the plane is determined (up to isometry) by the set of isometry types of its vertex-deleted subpolygons. In the final Section 4 a few open problems are mentioned.
2.
THE
MAIN
TOOLS
In this section and throughout the paper we use the following standard notation from permutation group theory. For a permutation group G acting on X, and for Y and S subsets of X, let YG = { g Y: g E G } denote the orbit of Y under the action of G, let G y = (g E G: gY = Y } denote the
155
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stabilizer of Y, and let YGs= { g Y: g E G, gS = S> denote the restricted orbit of Y under the action of G,. We start with the following somewhat technical but useful theorem. THEOREM 2.1. Let X be a (finite or infinite) set and let G be a (finite or infinite) group of automorphisms of X. Let Y be a subset of cardinality m of X, which is not reconstructible from its k-deck. Suppose, further, that there is a subset S of Y of cardinality ( S( = t, with a finite stabilizer (G, ( < W, where m -k 2 t. Then there is a set T of cardinality 1T 1>/ m - k + 1, satisfying S c T z Y, and there is an EE (0, I> such that for every set K satisfying SGKETandjK/=&(mod2), thereisagEGso that TngY=K.
Proof Since Y is not reconstructible from its k-deck there is another subset Y’ c X, Y’ +! YG having the same k-deck as Y (and hence, clearly, 1 Y’ 1= m). For any set A containing S, let f,(A) denote the number of members of YG that contain A. (Notice that the condition 1Gs 1< cc easily implies that f 1(S) < co and hence, also, f,(A ) < a for all S c A.) Similarly, let f*(A) denote the number of members of Y’G that contain A. Define also f = fi - f2. One can easily check that for every set A containing S whose cardinality is at most m -k, f,(A) can be determined from the k-deck of Y. (In fact:f,(A)=(l/(“,‘Al)) CrEdyk 1(gEG:A cgr)l.) Sincef@) is determined from d Y’,k= dy,k in the same manner we conclude that forall
fU)=fM-f,(A)=0
ScA,
IAl 1 #O. Also, by (2.1), ) TI am-k+ 1. Put f(T)=f,(T)-f,(T)=b. Let K be a set satisfying S c KG T. By a simple inclusionexclusion on the number of sets gY containing K we obtain I(gEG:gYnT=K}I
=
c
(- l)‘K’-K’
f,(K’).
(- l)‘K-K’
f2(K’).
K’; KcK’cT
Similarly I{gEG:gY’nT=K)I=
1 K’;KcK’cT
By subtracting
the second equality from the first we conclude
I(gEG:gYnT=K)I--I{gEG:gY’nT=K}I =
c
(-l)‘K’-K’f(K’)=(_l)‘T--KI
.b,
K’:KEK’GT
where the last equality follows from the minimality
582bj47/2-3
in the choice of T.
156
ALONETAL.
We conclude that there is an EE (0, 1 } such that for every S s Kc T with llyl =s(mod 2) there is a ge G so that gYn T= K. (And also, for every ScKcT with IKI-(l-s)(mod2) there is a gEG so that gY'n
T= K.)
This completes the proof.
1
Remark 2.2. An equivalent formulation of the last proof is related to the study of null q-designs and to the Reed-Muller codes (see [AL, FP, MS). A real function h defined on the power set of a set X is a null q-design if for every A g X of cardinality I A I < q the equality C&Ad h(B) = 0 holds. One can easily check that if in Theorem 2.1, t = 0, then the function h defined on the power set of X by h(A) = 1 if A E p, h(A) = - 1 if A E YIG, and h(A) = 0 otherwise is a null (m - k)-design. In fact, Theorem 2.1 for this case can be derived from the main result of [FP]. Null q-designs have recently been generalized to arbitrary ranked finite lattices in [Le] where the ideas from [FP] are extended using the theory of Mobius functions. It is possible to obtain a version of Theorem 2.1 that deals with general ranked finite lattices, but for our purposes here the present formulation suffices. Remark 2.3. Theorem 2.1 with X being the set of edges of the complete graph K,, on rz vertices, G being the group of its automorphisms, S = 0, t = 0, and k = 1 is just the result of Nash-Williams [N] concerning the edge reconstruction conjecture. COROLLARY 2.4. Let G be a finite group acting on X and let Y be a subset of cardinality m of X. If 2”- k > )G l/l G y I then Y is reconstructible from its k-deck. In particular, tf 2”- k > I G I then every subset of cardinality m of X is reconstructible from its k-deck.
Proof Clearly I YG I = ( G l/l GY I. If Y is not reconstructible from its k-deck, then, by Theorem 2.1 (with S= 0) there is a subset T of cardinality 1T I > m - k + 1 of Y that has at least 2’ ‘I - ’ > 2m-k distinct intersections with members of YG. Thus I G l/l GY I > 2m-k. This contradicts the assumption and hence Y is reconstructible, as required. 1 COROLLARY 2.5. Let G be a group acting on X and suppose Y~X,IYI=m.Suppose,further, that S~Y,ISI=t,(Gsl I YesI . (7) then Y is reconstructible from its k-deck.
Proof Put I YGSI = s. Suppose the assertion of the corollary is false and Y is not reconstructible from its k-deck. By Theorem 2.1 there is a subset T, Ss Ts Y, of cardinality I TI am-k+ 1 and an aE (0, l> such that for every K, SC KG T, I K( -E (mod 2), there is a gEG so that TngY= K. Put 0={gY:g~G, ScgY}. Let Y(t) be the set of all subsets of car-
RECONSTRUCTIONPROBLEMS
157
dinality t of Y. Clearly 1Y(t)/=(y) and O=UzEY(IJ {gY:gEG,S=gZ}. One can easily check that for each fixed 2, 1{g Y; g E G, S = gZ} 16 I YGSl= s. Hence ( 0 / 4; i.e., every graph with n vertices and m 2 4 edges is reconstructible from its edgedeleted subgraphs. Corollary 2.5 (with t = 1) gives that if 2m-2 > m -2. (n - 2)! then any graph with n vertices and m edges is reconstructible from its edge-deleted subgraphs. This is (slightly) better than Muller’s result [M], which follows, of course, from Corollary 2.4. Another result which follows from Theorem 2.1 is the following. PROPOSITION 3.1. Let H be a connected graph with n vertices, m edges, and maximum degree A. If
2m-k-“+1>n.A!(A-1)“-d-1
P-1 )
then H is reconstructible from the collection of its k-edge deleted subgraphs.
We note that this implies that if d 2 2 + 2 log, A + 2(k - 1 )/n, where d is the average degree in H, then H is reconstructible from its k-edge deleted subgraphs. This extends a theorem of Caunter and Nash-Williams (cf. [B] ), who proved the case k = 1 of the above statement. ProoJ Suppose the proposition is false and let H be a counterexample. Let Y be the set of edges of H and let S be the set of edges of an arbitrary spanning tree with maximum degree A in H. Since Y is not reconstructible from its k-deck, Theorem 2.1 implies that there is a set of edges T, SETG Y, ITI am-k+ 1, and an EE (0, 11, such that for every K, S c Kc T, I K I - E (mod 2), there is an automorphism g of K,, such that gYn T= K. Hence I { gY: ScgY} I b 21T’-IsI-1 2 2m-k-n+1. However, the set { gY: S cgY} is the number of distinct copies of the graph H that contain the edges of the fixed spanning tree S. It is not too difficult to show
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ET AL.
that this number does not exceed ~1.d!(d - l)“-d-l (see, e.g., [B]). Hence, 2+‘-“+’ < nA!(A - l)n--d- ‘, contradicting inequality (3.1). Therefore, H is reconstructible and the assertion of Proposition 3.1 holds. 1 Similar results hold for uniform or non-uniform hypergraphs. Here X is the set of all 2” edges of the complete hypergraph H, on n labelled vertices and G is the group of n! automorphisms of H,. Corollary 2.4 implies here that any hypergraph with n vertices and WI edges is reconstructible from its edge-deleted subgraphs, provided 2”- ’ > n!. Analogous results for reconstruction from k-edge-deleted subgraphs can be obviously formulated and proved, as well. Let us mention an application of the last result to the vertex reconstruction conjecture, that asserts that any graph with n 2 3 vertices is reconstructible from its vertex-deleted subgraphs. This conjecture is not a special case of Problem 1.1, but for certain graphs it is. Indeed, let G = ( V, E) be a 2-connected bipartite graph with classes of vertices A and B, where IBI bl +log,(lA I!) an d in which each two vertices in B have distinct sets of neighbours. Associate each such graph G with a hypergraph H(G) on the set of vertices A whose I B I edges are the sets eb = {a E A; ab E E} (b E B). One can easily check that for every b E B, H( G\b) = H(G) - eb and hence, the vertex-reconstructibility of G follows from the edge-reconstructibility of the hypergraph H(G). 3.2. The Cycle Problem and Its Extension
Let X be a cycle of length n and let Y be a coloring of its vertices by two colors, denoted 0 and 1. Suppose m vertices are colored 1 and suppose 1 < k < m. If we are given the collection of all (y )( unlabelled) colored cycles obtained from Y be replacing k l’s by O’s, can we reconstruct Y (up to rotation and reflection)? This problem is obviously a special case of the general Problem 1.1 presented in Section 1. Here X is the set of n vertices of the cycle and G is the group of its automorphisms, i.e., the dihedral group of order 2n. By Corollary 2.4, if m > log,(2n) + k then Y is reconstructible. However, Corollary 2.5 is more effective here. Clearly, for every x E X, (G, I = 2. Hence, by Corollary 2.5 if m > log,(2m)
+ k+ 1
(3.2)
then Y is reconstructible. For k = 1 this shows that Y is reconstructible for all m 2 6. We note that for k = m - 2 there are non-reconstructible colorings with larger values of m. To see this recall that a difference set modulo n is a subset D E 2, so that every 0 # x E 2, can be expressed uniquely as a difference d- d’ where d, d’ ED. Let D, and D, be two difference sets modulo and D,# -D,+x for every XEZ,. Put n, so that D,#D,+x I D1 I = ID, I = m. Identify the n vertices of the cycle X with the elements of
RECONSTRUCTION
159
PROBLEMS
2, and define two colorings Y, and Y2 of it by Y,(x) = 1 iff x E Di (i = 1,2). One can easily check that these two colorings are not isomorphic, but the two collections of all (7) colorings obtained from each of them by replacing in all possible ways m - 2 l’s by O’s are identical. As an example take n = 13, m = 4, k = 2, D, = (0, 1,4, 6}, and D, = (0, 2, 8, 12). This shows that for m = 4 and k = 2 not every Y is reconstructible. By inequality (3.2) for k = 2 if m 2 7 then Y is reconstructible. The cycle problem can be generalized to arbitrary graphs. Let H be a graph on a set X of vertices, and let Y be a coloring of its vertices by the two colors 0 and 1, with m vertices colored 1. Suppose we are given the collection of all ( y) (unlabelled) colorings obtained from Y by replacing in all possible ways k l’s by 0’s. Can we then reconstruct Y? Here G is the group Aut H of automorphisms of H. Thus, e.g., if 2” - ’ > 1Aut H 1 then Y is reconstructible by Corollary 2.4. The cycle problem is an interesting example, since here, by Corollary 2.5, our results do not depend on the length n of the cycle. For every fixed k, any coloring with sufficiently large m is reconstructible. There are many other examples of such graphs H, e.g., the product of an n-cycle with an edge, or with a triangle. Similar results hold for edge coloring, where the edge reconstruction conjecture is an instance of this last problem, with H = K, and k = 1. 3.3. Geometrical
Reconstructions
In the previous applications, the automorphism group G considered was always finite. Here we mention some infinite examples. Let X be a subset of the 2-dimensional Euclidean space lw2 (we will be interested in the cases X=R2 or X=S’={( x,,x,)~R~:x:+x:=l}). Let G be the group of all isometries of X. Thus, for X= Iw2 or X= S’, two subsets of X are G-equivalent iff they are isometric. If Y is a finite subset of cardinality m of X and k cm, then the k-deck of Y is the set of all (T) isometry types of all the subsets of cardinality m-k of Y, each appearing according to its multiplicity. We fn-st consider the case X = S’. Any finite subset of X is the set of vertices of a polygon, inscribed in a unit circle. Clearly, here, for every x E X 1G, 1= 2 and hence, by Corollary 2.5, if m > k + 1 + log,(2m) then Y is reconstructible from its k-deck. Notice that the example given in the previous section is an example of a set of four points which is not reconstructible from its 2-deck. For subsets of S’ we can show that “similar” k-decks correspond to “similar” subsets. More formally, let us define the distance d( Y, Y’) between two finite subsets of the same cardinality Y and Y’ of S’ as 1 dist(gy,f(y)):gEG,f: y
Y+ Y’abijection
YE
where dist( gy,f( y)) is the usual (Euclidean)
distance between gy andf( y).
160 In this notation THEOREM 3.1. and suppose that dinality m - k of so that for every
ALONETAL.
we prove: Suppose Y, Y’ c S’, 1Y 1= 1 Y’I = m > k + 1 + log,(2m), there is a bijection h from the set F of all (7) subsets of carY to the set F’ of all (r) subsets of cardinality m - k of Y’ TE F, d( T, h(T)) < E< 1/202”+ ‘. Then d( Y, Y’) < 40 .rn -1.
Proof The points of S’ are just the points {e2rrir: 0 < x < 11. By the well-known result of Kronecker on simultaneous approximations (see, e.g., [HW]) there is an integer p, l/20&
4m is enough, as the maximaum distance between a pair of m points in the plane is obtained at most m times (see, e.g., [E] ). Thus, every set of m > 8 points in [w2 is uniquely determined (up to isometry) by the isometry types of all its point-deleted subsets. For some related results, see [KR3]. The above results can be generalized to sets of points in general position in higher dimensions. We omit the details. 4. OPEN PROBLEMS The most interesting problem is of course the edge reconstruction conjecture. It does not seem that the general approach considered here will be very fruitful in tackling this problem; one will probably have to use more specific combinatorial arguments. In the cycle problem, considered in Section 3, it would be nice to decide
RECONSTRUCTION
PROBLEMS
161
precisely for every pair m and k if every cycle coloring with m l’s can be reconstructed from the collection of the (T) colorings obtained from it by replacing k l’s by O’s in all possible ways. Our results suggest that the size n of the colored cycle is not crucial here. A similar question arises naturally for the geometric problem. It would be interesting to decide, for every m and k, if every set of m points in R2 is determined (up to isometry) by its k-deck.
REFERENCES
[AL]
WI [El [FP] [GKR]
WI
N. ALON AND A. LIU, An application of set theory to error correcting codes, Math. Msg., in press. J. A. BONDY, The reconstruction of graphs, preprint, 1984. P. ERD&, On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248-250. P. FRANKL AND J. PACH, On the number of sets in a null t-design, European J. Combin. 4 (1983), 21-23. C. D. GODSIL, I. KRASIKOV, AND Y. RODITTY, Reconstructing graphs from their k-edge deleted subgraphs, J. Combin. Theory Ser. B 43 (1987), 360-363. F. HARARY, On the reconstruction of a graph from a collection of subgraphs, in
“Theory of Graphs and its Applications” (M. Fiedler, Ed.), pp. 47-52, Czechos. Acad. Sci. Prague, 1964. [HW] G. H. HARDY AND E. M. WRIGHT, “An introduction to the Theory of Numbers,” Oxford, 1945. [KRl ] I. KRASIKOV AND Y. RODITY, Balance equations for reconstruction problems, Arch. Math. 48 (1987) 458-464. [KR2] I. KRASIKOV AND Y. RODITTY, A counting technique for reconstruction problems, Ars Combin. in press. [KR3] I. KRASIKOV AND Y. RODITTY, Geometrical reconstruction. Ars Combin., in press. CL1 L. LovAsz, A note on the line reconstruction problem, J. Combin. Theory Ser. B 13 (1972), 309-3 10. H. LEFMANN, An extremal problem for Graham-Rothschild parameter words, WeI Combinatorics, in press. [MS] F. J. MCWILLIAMS AND N. J. A. SLOANE, “The Theory of Error Correcting Codes,” North-Holland, Amsterdam, 1977. CM1 V. MULLER, The edge reconstruction conjecture is true for graphs with more than n log, n edges, J. Combin. Theory Ser. B 22 (1977), 281-283. C. ST. J. A. NASH-WILLIAMS, The reconstruction problem, in “Selected Topics in CW Graph Theory” (L. W. Beineke and R. J. Wilson, Eds.), pp. 205-236, Academic Press, San Diego, 1978.
